A scatter diagram shows whether two measured variables are associated, and how strongly. A line of best fit lets you predict one value from the other, but the diagram cannot prove that one variable causes the other.
This skill belongs to data displays and cumulative reasoning. Exam questions usually ask you to describe the correlation, draw or use a line of best fit, and comment on a conclusion.
How do you describe what a scatter diagram shows?
Use two words: direction and strength. Positive correlation means that as x increases, y tends to increase.
Negative means y tends to decrease. No correlation means there is no clear pattern.
Strong means the points lie close to a straight line, and weak means they are spread out. Always describe it in the context, for example “students who revised longer tended to score higher”, not just “positive”.
Drawing and using a line of best fit, step by step
- Plot every point carefully. Circle any point that sits far from the pattern, called an outlier.
- Draw a straight line that follows the trend, with roughly equal numbers of points above and below it. It should pass close to the mean point if you know it.
- Read a prediction by going up from the x-value to the line, then across to the y-value.
- Check the range. A prediction inside the plotted x-values is interpolation. A prediction outside is extrapolation and is unreliable.
- Write the conclusion carefully: the variables are associated, and any suggestion of a reason needs more evidence.
Worked example
Eight students recorded the hours they revised for a test and their scores (out of 100).
| Hours revised x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Score y | 35 | 42 | 48 | 55 | 58 | 66 | 70 | 75 |
(a) Describe the correlation. The scores rise as the hours rise, and the points lie close to a line. It is strong positive correlation.
(b) A line of best fit passes through (2, 42) and (6, 66). Find its equation.
Gradient = (66 − 42) ÷ (6 − 2) = 24 ÷ 4 = 6. Then y = 6x + c. Using (2, 42): 42 = 12 + c, so c = 30.
The line is y = 6x + 30. Check with (6, 66): 6 × 6 + 30 = 66. ✓
(c) Interpret the gradient. A gradient of 6 means that each extra hour of revision is associated with about 6 more marks.
(d) Predict the score for 5.5 hours. y = 6 × 5.5 + 30 = 33 + 30 = 63. This is interpolation, since 5.5 is between 1 and 8, so it is reasonably reliable.
(e) Predict the score for 15 hours. The equation gives 6 × 15 + 30 = 120, which is above the maximum score of 100. This is extrapolation and the answer is not sensible.
The mistake to watch for
The usual slip is turning association into a cause.
Mistaken conclusion: “Ice-cream sales and sunburn cases are strongly correlated, so ice cream causes sunburn.”
The student treated the pattern as a mechanism. Hot, sunny weather drives both, so neither causes the other.
The correction is to name a possible third factor, here the weather, and to say “are associated”. Another classic example is children’s shoe size and vocabulary: both increase with age, but bigger shoes do not teach more words.
Check yourself
Try these, then open each answer.
1. Using y = 6x + 30 from the worked example, predict the score for 3.5 hours of revision.
Show answer
y = 6 × 3.5 + 30 = 21 + 30 = 51. This is interpolation because 3.5 lies within the data.
2. Why is the prediction for 12 hours not reliable?
Show answer
The equation gives 6 × 12 + 30 = 102, which is more than the maximum mark of 100, and 12 hours is outside the data (1 to 8 hours). It is extrapolation, so the pattern may not continue.
3. A café plots outdoor temperature against the number of hot drinks sold each day. What type of correlation would you expect, and can the café conclude cooler weather causes people to want hot drinks?
Show answer
Negative correlation: hot drink sales rise as the temperature falls. The scatter diagram shows association. It does not prove cause on its own, although a sensible reason is plausible, so the careful wording is “is associated with”.
Where this leads next
Scatter diagrams often sit beside other displays in a question, so practise choosing a display that preserves the data meaning. For comparing groups rather than variables, continue with comparing two distributions.
Then try the data displays practice set. The non-calculator working trainer helps with the gradient arithmetic.
Some students can compute a line and still write a conclusion that claims too much. Seeing your wording and adjusting it is something a teacher does well in online one-to-one Mathematics tuition.